Algebra Examples

$f (x) = 8 x^{6} - 64 x^{5} - 640 x^{2} - 512 + 1024 x + 160 x^{4}$

Step 1

Identify the degree of the function.

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Step 1.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$8 x^{6} \to 6$

$- 64 x^{5} \to 5$

$- 640 x^{2} \to 2$

$- 512 \to 0$

$1024 x \to 1$

$160 x^{4} \to 4$

Step 1.2

The largest exponent is the degree of the polynomial.

$6$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Move $- 512$ .

$8 x^{6} - 64 x^{5} - 640 x^{2} + 1024 x + 160 x^{4} - 512$

Step 3.1.2

Move $1024 x$ .

$8 x^{6} - 64 x^{5} - 640 x^{2} + 160 x^{4} + 1024 x - 512$

Step 3.1.3

Move $- 640 x^{2}$ .

$8 x^{6} - 64 x^{5} + 160 x^{4} - 640 x^{2} + 1024 x - 512$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

$8 x^{6}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

$8$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and rises to the right

Step 7